Properties

Label 2-300-1.1-c5-0-14
Degree $2$
Conductor $300$
Sign $-1$
Analytic cond. $48.1151$
Root an. cond. $6.93650$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 16·7-s + 81·9-s − 564·11-s + 370·13-s + 1.08e3·17-s − 2.86e3·19-s + 144·21-s − 1.58e3·23-s + 729·27-s + 1.13e3·29-s − 6.01e3·31-s − 5.07e3·33-s + 538·37-s + 3.33e3·39-s + 1.13e4·41-s − 5.44e3·43-s − 1.02e4·47-s − 1.65e4·49-s + 9.77e3·51-s − 3.47e4·53-s − 2.57e4·57-s − 2.61e4·59-s + 9.42e3·61-s + 1.29e3·63-s + 5.11e4·67-s − 1.42e4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.123·7-s + 1/3·9-s − 1.40·11-s + 0.607·13-s + 0.911·17-s − 1.81·19-s + 0.0712·21-s − 0.624·23-s + 0.192·27-s + 0.250·29-s − 1.12·31-s − 0.811·33-s + 0.0646·37-s + 0.350·39-s + 1.05·41-s − 0.449·43-s − 0.679·47-s − 0.984·49-s + 0.526·51-s − 1.69·53-s − 1.04·57-s − 0.979·59-s + 0.324·61-s + 0.0411·63-s + 1.39·67-s − 0.360·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(48.1151\)
Root analytic conductor: \(6.93650\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 300,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p^{2} T \)
5 \( 1 \)
good7 \( 1 - 16 T + p^{5} T^{2} \)
11 \( 1 + 564 T + p^{5} T^{2} \)
13 \( 1 - 370 T + p^{5} T^{2} \)
17 \( 1 - 1086 T + p^{5} T^{2} \)
19 \( 1 + 2860 T + p^{5} T^{2} \)
23 \( 1 + 1584 T + p^{5} T^{2} \)
29 \( 1 - 1134 T + p^{5} T^{2} \)
31 \( 1 + 6016 T + p^{5} T^{2} \)
37 \( 1 - 538 T + p^{5} T^{2} \)
41 \( 1 - 11370 T + p^{5} T^{2} \)
43 \( 1 + 5444 T + p^{5} T^{2} \)
47 \( 1 + 10296 T + p^{5} T^{2} \)
53 \( 1 + 34758 T + p^{5} T^{2} \)
59 \( 1 + 444 p T + p^{5} T^{2} \)
61 \( 1 - 9422 T + p^{5} T^{2} \)
67 \( 1 - 51124 T + p^{5} T^{2} \)
71 \( 1 - 14520 T + p^{5} T^{2} \)
73 \( 1 - 22678 T + p^{5} T^{2} \)
79 \( 1 + 97312 T + p^{5} T^{2} \)
83 \( 1 - 7956 T + p^{5} T^{2} \)
89 \( 1 + 47910 T + p^{5} T^{2} \)
97 \( 1 + 140738 T + p^{5} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.45510871927550969046411536354, −9.494300790770024996409062719922, −8.295060598907612078781866727664, −7.85591416078633649898367341606, −6.50672061536840876979896588173, −5.37126126972009857473818381797, −4.14084738349468823094220339644, −2.92677713764474412816139592562, −1.75312656472247586261929350791, 0, 1.75312656472247586261929350791, 2.92677713764474412816139592562, 4.14084738349468823094220339644, 5.37126126972009857473818381797, 6.50672061536840876979896588173, 7.85591416078633649898367341606, 8.295060598907612078781866727664, 9.494300790770024996409062719922, 10.45510871927550969046411536354

Graph of the $Z$-function along the critical line